Arithmetic sequences with fractions

Learning Objectives Identify if a sequence of fractional angles forms an arithmetic sequence. Calculate the common difference (as a fraction) in an arithmetic sequence of angles, including those in radian measure. Determine the nth term (a specific angle) in an arithmetic sequence using the formula a_n = a_1 + (n-1)d. Calculate the sum of a finite arithmetic sequence of fractional angles, such as the interior angles of a polygon. Solve for the first term, common difference, or number of terms in problems involving angles in polygons. Apply the concept of arithmetic sequences to rotational angles on the unit circle using radian measures. Ever noticed how a spiral staircase is built from steps that turn at the exact same angle each time? 📐 That's an arithmetic sequence i...

TermDefinitionExample Arithmetic SequenceA sequence of numbers where the difference between consecutive terms is constant. In this context, the 'numbers' are angle measures.The sequence of angles 90°, 97.5°, 105°, 112.5°, ... is an arithmetic sequence because you add 7.5° (or 15/2°) each time. Common Difference (d)The constant fractional amount added to each term to get the next term. It can be positive or negative.In the sequence of radian measures π/2, π/3, π/6, 0, ..., the common difference is d = π/3 - π/2 = 2π/6 - 3π/6 = -π/6. Term (a_n)An individual angle measure in a sequence. a_1 is the first angle, a_n is the nth angle.In the sequence 100°, 104.5°, 109°, ..., the third term is a_3 = 109°. Interior Angles of a PolygonThe angles on the inside of a polygon formed by adjace...

nth Term of an Arithmetic Sequence a_n = a_1 + (n-1)d Use this formula to find the measure of a specific angle (a_n) in a sequence, given the first angle (a_1), its position in the sequence (n), and the common difference (d). Sum of a Finite Arithmetic Sequence S_n = n/2 * (a_1 + a_n) Use this formula to find the sum of the first n angles in a sequence when you know the first angle (a_1), the last angle (a_n), and the number of angles (n). Alternative Sum Formula S_n = n/2 * (2a_1 + (n-1)d) Use this when you need to find the sum of angles but do not know the last angle (a_n). It is particularly useful for solving for an unknown variable like 'd' or 'a_1'. Sum of Interior Angles of a Polygon Sum = (n-2) * 180° A fundamental geometric rule t...